1. Mathematics
Quarter 3 - Module 5
Solving Problems Involving
Permutations and Combinations
Department of Education ● Republic of the Philippines
2. Mathematics - Grade 10
Alternative Delivery Mode
Quarter 3 - Module 5: Solving Problems Involving
Permutations and Combinations
First Edition, 2020
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Published by the Department of Education
Secretary: Leonor Magtulis Briones
Under Secretary: Diosdado M. San Antonio
Development Team of the Module
Author/s: Decemay D. Labadan, SST I
Reviewers: Elbert R. Francisco, PhD
Rhodel A. Lamban, PhD
Jasmin G. Sulatan
Illustrator and Layout Artist: Decemay D. Labadan
Management Team
Chairperson: Arturo B. Bayocot, PhD, CESO III
Regional Director
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Schools Division Superintendent
Shambaeh A. Usman, PhD
Assistant Schools Division Superintendent
Mala Epra B. Magnaong, Chief, CLMD
Neil A. Improgo, PhD, EPS-LRMS
Bienvenido U. Tagolimot Jr., PhD, EPS-ADM
Members Elbert R. Francisco, PhD, Chief, CID
Rhodel A. Lamban, PhD, EPS Mathematics
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Jeny B. Timbal, PDO II
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3. Mathematics
Quarter 3 - Module 5
Solving Problems Involving Permutations
and Combinations
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10
4. Table of Contents
Page
WHAT THIS MODULE IS ABOUT
Note to the Teacher/Facilitator
Note to the Learner
Note to the Parents/Guardian
Module Icons
WHAT I NEED TO KNOW
WHAT I KNOW (Pretest)
Lesson 1:
Permutation of n taken r at a time 2
What I Need to Know 2
What I Know 3
What’s In 4
What’s New 4
What Is It 5
What’s More 5
What I Have Learned 5 - 6
What I Can Do 6
Assessment 6 - 7
Lesson 2:
Permutation of n distinct objects arranged in a circle 7
What I Need to Know 7
What I Know 8 - 9
What’s In 9
What’s New 9
What Is It 10
What’s More 10
What I Have Learned 10
What I Can Do 11
Assessment 11 - 12
Lesson 3:
Distinguishable Permutation 12
What I Need to Know 12
What I Know 13
What’s In 14
What’s New 14
What Is It 15
What’s More 15
What I Have Learned 16
What I Can Do 16
Assessment 16 - 17
5. Lesson 4:
Combination of n taken r at a time 17
What I Need to Know 17
What I Know 17 - 18
What’s In 19
What’s New 19
What Is It 20
What’s More 20
What I Have Learned 21
What I Can Do 21
Assessment 21 - 22
Summary 23
Assessment: (Post-Test) 24
Key to Answers 25
References 26
6. What This Module is About
In this module, you will learn more about problems involving
permutations and combinations. This will require the knowledge and skilld of
the basic counting techniques. It is an important skill that you need to develop
because it helps you a lot in doing the activities enjoyable.
Notes to the Teacher:
Welcome to the new normal way of teaching How to Solve Problems Involving
Permutations and Combinations through Alternative Delivery Mode (ADM).
As a teacher / facilatator, you play an importatnt role in your students
independent learning skills to be develop. You must be aware of your
students’ available learning materials at home such as electronic gadgets,
e.g. cell phones or personal computers and internet connection so you would
know the suitable learning activities for your students. You need to respond to
your learners with different interests and skills. You also need to be
accomodating to help facilitate learning opportunities.
Notes to the Learner:
Welcome to the new normal way of learning How to Solve Problems Involving
Permutations and Combinations.
This module is designed for you to be prepared with the essential
competencies needed. Manage your time well so that you would be able to
complete this course. If some of the exercises are difficult to undrestand, do
not give up. Please ask help from your teacher or anyone whom you know
that can help you. Enjoy this distinct and meaningful learning experience.
Your success lies in you!
Notes to the Parent / Guardian:
As we adopt the new normal way of learning, blended education is our
alternative to face to face classes. We know that the kind of education we will
give to your child/children abides to the vision and mission of the Department
of Education. Thus, it is necessary to work together to achieve the quality of
education your child/children deserve.
7. Module Icons
What I Need to
Know
This part contains
learning objectives that
are set for you to learn
as you go along the
module each day/lesson.
What I know
This is a pre-test
assessment as to your
level of knowledge to the
subject matter at hand,
meant specifically to
gauge prior related
knowledge.
What’s In
This part connects
previous lesson with that
of the current one.
What’s New
An introduction of the
new lesson through
various activities, before
it will be presented
to you.
What is It
These are discussions of
the activities as a
way to deepen your
discovery and under-
standing of the concept.
What’s More
These are follow-up
activities that are in-
tended for you to practice
further in order to master
the competencies.
What I Have
Learned
Activities designed to
process what you have
learned from the lesson.
What I can do
These are tasks that are
designed to show-case
your skills and
knowledge gained, and
applied into real-life
concerns and situations.
8. What I Need to Know
In this lesson, you will:
1. Evaluate the factorial of a number.
2. Derive the formula for finding the number of permutations of n objects
taken r at a time.
3. Solve problems involving permutation.
4. Illustrate the combination of objects.
5. Derive the formula for finding the number of combinations of n objects
taken r at a time.
6. Solve problems involving combinations.
What I Know
PRE-TEST
Find out how much you already know about this topics in this module. Choose
the letter of the best answer. Take note of the items that you were not able to
answer correctly and find out the right answer as you go through this module.
1. It is a way of selecting a subset from the given set where order does
not matter.
A. Combination C. Permutation
B. Integration D. Simulation
2. In how many ways can you arrange 9 pots in a row?
A. 40 320 B. 42 300 C. 362 880 D. 380 620
3. Find the number of distinguishable permutations of the letters of the
word COMMITTEE.
A. 9 B. 18 C. 40 320 D. 45 360
4. In how many ways can 10 people be seated around a circular table?
A. 368 028 B. 362 880 C. 803 268 D. 862 286
5. What is P(9,2)?
A. 2 B. 72 C. 40 320 D. 362 880
6. In a room, there are 8 chairs in a row. In how many ways can 6
students be seated in consecutive chairs?
A. 2 B. 720 C. 20 160 D. 21 060
7. What is P(8, 3)?
A. 6 B. 336 C. 633 D. 40 320
8. In how many ways can 10 students arrange themselves in a
picture taking?
A. 8 362 300 B. 3 628 800 C. 800 632 D. 362 800
9. Find: C(9, 3)
A. 6 B. 84 C. 720 D. 362 880
1
9. 10.A restaurant offers 4 kinds of soup, 7 kinds of main dish, 5 kinds of
vegetable dish, and 6 kinds of dessert. In how many possible ways can
the restaurant form a meal consisting of 1 soup, 2 main dish, 3
vegetable dish, and 2 desserts?
A. 8 B. 22 C. 106 D. 990
11. Solve : C(10, 7) = _____.
A. 25 B. 45 C. 100 D.120
12.In how many ways can a committee of 9 students be chosen from 8
juniors and 10 seniors if there must be 5 seniors in the committee?
A. 190 B.920 C. 30 190 D. 31 920
13.Find: C(8, 3)
A. 36 B. 46 C. 56 D. 66
14. In how many ways can n different objects be arranged taken all at a
time?
A. (n – 1)! B. (n + 1)! C. n! D. (2n)!
15. What is the arrangement of finite numbers of objects taken some or all
at a time?
A. Combination B. Permutation C. Probability D. Statistics
Lesson
1
What I Need to Know
This lesson is written for you to further understand the concept of solving
problems involving permutations using the basic counting techniques. Since
permutation is an ordered arrangement of either all objects in a set or part of
such set, so we can say that the order of each elements is important. To solve
for permutation of n taken r at a time, use the permutation formula:
nPr = n!__
(n – r)!
2
Permutation of n taken r at
a time
10. What I Know
Pretest
As part of your initial activity, you need to answer this pre-assessment
to check your prior knowledge on the topics of this module.
Direction: Read carefully each item below. Write all your answer on your
activity notebook.
Find the value of the following:
1. 8! 4. 5! + 9!
2. 4! . 6! 5. 12!
3!
3. 10P3
Solve each using the permutation formula:
6. In how many ways can you arrange 7 photos in a row?
A. 702 B. 720 C. 5004 D. 5040
7. Joshua wants to visit 5 resorts. In how many ways can he make the trip
to the resorts?
A. 24 B. 42 C. 102 D. 120
8. You are asked to choose best 6 out of 10 songs. In how many ways
can you choose the top 6 songs?
A. 151 B. 201 C. 150 201 D. 151 200
9. How many 3-digit number can be formed from the digits 3, 4, 5, 6, 7
and 8? A. 6 B. 60 C. 102 D. 120
10.In how many ways can 8 girls sit if there are only 5 chairs available?
A. 120 B. 6227 C. 6720 D. 40 320
11. In how many different ways can the letters of the word “WARPING” be
arranged in such a way that all letters always come together?
A. 360 B. 480 C. 720 D. 5040
12. What is P(9, 2)?
A. 88 B. 72 C. 50 D. 42
13. What is P(12, 5)?
A. 120 B. 5040 C. 95 040 D. 479 0001 600
14.Which of the following is equal to P(6, 4)?
A. 2 B. 360 C. 720 D. 823
15.Which of the following is equal to P(9, 9)?
A. 1 B. 360 C. 360 800 D. 362 880
3
11. What’s In
In this lesson, the basic cooncept of operation is very much needed. Not only
the mastery of the operation but also you need to understand the problem set
well and the appropriate formula to be used.
What’s New
Let’s have another activity, have fun and enjoy!
Activity 1: Let’s Find Out!
Evuate the following:
A. 5P2 C. 8P4
B. 9P7 D. 6P3
Solution:
A. 5P2 = 5!__ C. 8P4 = 8!__
(5 – 2)! (8 – 4)!
= 5! = 8!
3! 4!
= 5.4.3.2.1 = 8.7.6.5.4.3.2.1
3.2.1 4.3.2.1
= 5.4 = 8.7.6.5
5P2 = 20 9P7 = 1,680
B. 9P7 = 9!_ D. 6P3 = 6!__
(9 – 7)! (6 – 3)!
= 9! = 6!
2! 3!
= 9.8.7.6.5.4.3.2.1 = 6.5.4.3.2.1
2.1 3.2.1
= 9.8.7.6.5.4.3 = 6.5.4
9P7 = 181, 440 6P3 = 120
4
12. What Is It
Activity 2: Count Me In!
1. In how many ways can 7 boys sit in a row of 5 chairs?
Solution:
The number of permutations of 7 objects taken 5 at a time is
7P5 = 7! = 7! = 7.6.5.4.3 = 2, 520 ways
(7 – 5)! 2!
Therefore 7 boys can sit in a row of 5 chairs in 2,520 ways.
2. In how many different ways can 6 motorcycles be parked if there are
10 parking spaces provided?
Solution:
The number of permutations of 10 objects taken 6 at a time is
10P6 = 10! = 10! = 10.9.8.7.6.5 = 151, 200 ways
(10 – 6)! 4!
Therefore 6 motorcycles be parked to 10 parking spaces in 2,520 ways.
What’s More
Activity 3: Find Me!
Solve for the number of possible outcomes. Write your solution in your
activity notebook:
1. P (9, 8)
2. P (6, 2)
3. P (5, 5)
4. P (10, 1)
5. P (4, 3)
What I Have Learned
Activity 4: Apply Your Skills
1. In how many ways can we arrange the letters of the word ROME
taken all at a time?
A. 4 ways B. 8 ways C. 12 ways D. 24 ways
5
13. 2. What is P(6,1)? A. 1 B. 6 C. 12 D. 18
3. What is P(8, 3)? A. 24 B. 83 C. 336 D. 512
4. Which of the following is equal to P(4, 3)?
A. 4/3 B. 12 C. 24 D. 43
5. Which of the following is equal to P(5, 2)?
A. 2/5 B. 5/2 C. 10 D. 20
Solve for the different permutations:
6. P (8, 2)
7. P (8, 8)
8. P (14, 2)
9. P (4, 4)
10.P (7, 5)
What I Can Do
Activity 5: A Journey Into the Unknown
Solve for the unknown in each item. Write your solution in your activity
notebook.
1. P(12,7)
A.3 991 680 B. 3 990 618 C. 3 680 991 D. 3 618 909
2. P(9, 4)
A. 3420 B. 3240 C. 3042 D. 3024
3. P(6,6)
A. 720 B. 702 C. 270 D. 207
4. P(4, 4)
A. 42 B. 24 C. 16 D. 6
5. P(10,9)
A. 3 828 600 B. 3 800 362 C. 3 800 328 D. 3 628 800
ASSESSMENT
Solve each using the permutation formula. Write your answer in your activity
notebook:
1. In how many different ways can the letters of the word “WARPING” be
arranged in such a way that all letters always come together?
A. 360 B. 480 C. 720 D. 5040
2. Joshua wants to visit 5 resorts. In how many ways can he make the trip
to the resorts?
A. 24 B. 42 C. 102 D. 120
6
14. 3. You are asked to choose best 6 out of 10 songs. In how many ways
can you choose the top 6 songs?
A. 151 B. 201 C. 150 201 D. 151 200
4. In how many ways can 8 girls sit if there are only 5 chairs available?
A. 120 B. 6227 C. 6720 D. 40 320
5. How many 3-digit number can be formed from the digits 3, 4, 5, 6, 7
and 8? A. 6 B. 60 C. 102 D. 120
6. Which of the following is equal to P(9, 9)?
A. 1 B. 360 C. 360 800 D. 362 880
7. What is P(9, 2)?
A. 42 B. 50 C. 72 D. 88
8. What is P(12, 5)?
A. 120 B. 5040 C. 95 040 D. 479 0001 600
9. In how many ways can you arrange 7 photos in a row?
A. 5004 B. 5040 C. 702 D. 720
10.Which of the following is equal to P(6, 4)?
A. 824 B. 720 C. 360 D. 2
Find the value of the following:
11. 5! + 9! 14. 10P3
12. 4! . 6! 15. 12!
13. 8! 3!
Congratulations! You are done with this lesson. I hope you have fun in
learning permutation. Good job!
Lesson
2
What I Need to Know
In this lesson, solving problems involving n distinct objects arranged in a circle
is well emphasized. To further understand the concept of solving problems
involving permutations, use the given formula as follow:
P = (n – 1)!
7
Permutation of n distinct
objects arranged in a circle
15. What I Know
Pre-test
Solve each using the circular permutation formula. Write your answers in your
activity notebook:
1. In how many ways can 5 plants be arranged in a spherical flower
stand? A. 24 B. 42 C. 102 D. 120
2. I a game called “Trip To Jerusalem”, in how many ways can 7 people
be seated?
A. 270 B. 720 C. 5004 D. 5040
3. In how many ways can 8 teachers be seated in a circular conference?
A. 5040 B. 5400 C. 32 400 D. 40 320
4. In a group of 9 teachers, in how many ways can they sit in a circular
meeting?
A. 40 302 B. 40 320 C. 360 882 D. 362 880
5. In a group of 7 children, in how many ways can they position
themselves in a round table?
A. 702 B. 720 C. 5 004 D. 5 040
6. How many possible ways can you arranged your 11 toys in a round
shelf?
A. 3 628 B. 3 800 C. 3 628 800 D. 3 682 080
7. How many arrangements can you form in a carousel with 12 children?
A. 39 800 B. 39 960 C. 39 961 080 D.39 916 800
8. How many possible ways can you position the 9 people in a ferris
wheel given 1 person per ferry?
A. 40 302 B. 40 320 C. 43 002 D. 43 200
9. In a game named “Sisira ang Bulaklak”, in how many ways can you
order the 5 kids in the game?
A. 24 B. 42 C. 102 D. 120
10. In a family of 7, in how many arrangement can be done during a
circular family dinner?
A. 207 B. 720 C. 5 004 D. 5 040
11. How many possible arrangements can be form given 10 persons in a
merry-go-round game?
A. 326 808 B. 362 880 C. 880 326 D. 880 362
12. In a circular permutation, how many ways can you arrange 4 objects at
a time? A. 6 B. 16 C. 24 D. 42
8
16. 13. In a round robin work, in how many ways can 15 utility workers
perform their job at a time?
A. 81 782 197 200 C. 87 121 920 087
B. 87 121 920 078 D. 87 178 291 200
14. In a rotational schedule, how many possible ways can 21 nurses be
arranged at a time?
A. 2.430 081 762 902 64 x 1018
B. 2.430 817 662 902 04 x 1018
C. 2.432 902 008 176 64 x 1018
D. 2.490 200 817 632 64 x 1018
15. How many ways can 14 students encircle the round table.
A. 6 002 270 200 C. 6 202 080 027
B. 6 020 800 227 D. 6 227 020 800
What’s In
This lesson focuses on the concepts underlying about CIRCULAR
PERMUTATIONS. As mention previously, the subtraction of 1 in the formula
accounts for the object that must be FIXED
What’s New
Let’s have another activity, have fun!
Activity 1:
Six people are going to sit a round table. How many different ways
can this be done?
Solution:
Let n = 6
Apply the formula:
P = (n – 1)!
P = (6 – 1)!
= 5!
P = 120 ways
Therefore, there are 120 different ways can be done when 6 people are going
to sit a round table.
9
17. What Is It
Activity 2: Find Out!
A couple wants to plant some shrubs around a circular walkway. They
have seven different shrubs. How many different ways can the shrubs be
planted?
Solution:
Let n = 7
Apply the formula:
P = (n – 1)!
P = (7 – 1)!
= 6!
P = 720 ways
Therefore, there are 720 different ways can the shrubs be planted
when a couple wants to plant some shrubs around a circular walkway.
What’s More
Activity 3: Analyze the problems deeply using circular permutations.
1. In how many ways can 9 people be seated in a Trip To Jerusalem?
2. In a circular conference,In how many ways can 13 students be seated?
3. In a group of 14 pupils, in how many ways can they sit in a circular
meeting?
What I Haved Learned
Activity 4: Evaluate the following using Circular Permutation.
1. How many possible ways can you arranged 7 kids in the game named
“Sisira ang Bulaklak”?
2. How many arrangement can be done during a circular family dinner In
a family of 16?
3. In a merry-go-round, how many possible arrangements can be form
given 10 persons?
10
18. What I Can Do
Activity 5: Examine the following problems using Circular Permutation.
1. How many ways can you arrange 6 objects at a time In a circular
permutation?
2. In a round robin work, in how many ways can 18 attendants perform
their job at a time?
3. In a group of 4 teens, in how many ways can they position
themselves in a spherical table?
4. How many possible ways can you arranged your 12 toys in a round
shelf?
Assessment
Solve each using the circular permutation formula. Write your answers in your
activity notebook:
1. How many ways can 14 students encircle the round table.
A. 6 020 800 227 C. 6 202 080 027
B. 6 082 270 200 D. 6 227 020 800
2. How many possible ways can you position the 9 people in a ferris
wheel given 1 person per ferry?
A. 40 302 B. 40 320 C. 43 002 D.43 200
3. In a group of 7 children, in how many ways can they position
themselves in a round table?
A. 702 B. 720 C. 5 004 D. 5 040
4. How many possible ways can you arranged your 11 toys in a round
shelf?
A. 3 628 B. 3 800 C. 3 628 800 D. 3 682 080
5. How many arrangements can you form in a carousel with 12 children?
A. 39 800 B. 39 960 C. 39 961 080 D.39 916 800
6. How many possible arrangements can be form given 10 persons in a
merry-go-round game?
A. 326 808 B. 362 880 C. 880 326 D. 880 362
7. In a circular permutation, how many ways can you arrange 4 objects at
a time? A. 6 B. 16 C. 24 D. 42
8. In a round robin work, in how many ways can 15 utility workers
perform their job at a time?
A. 81 782 197 200 C. 87 121 920 807
B. 87 121 920 078 D. 87 178 291 200
9. In a game named “Sisira ang Bulaklak”, in how many ways can you
order the 5 kids in the game?
A. 24 B. 42 C. 102 D. 120
11
19. 10. In a family of 7, in how many arrangement can be done during a
circular family dinner?
A. 207 B. 720 C. 5 004 D. 5 040
11.In how many ways can 5 people be seated in a round table?
A. 24 B. 42 C. 102 D. 120
12.In a rotational schedule, how many possible ways can 21 nurses be
arranged at a time?
A. 2.430 081 762 902 64 x 1018
B. 2.430 817 662 902 04 x 1018
C. 2.432 902 008 176 64 x 1018
D. 2.490 200 817 632 64 x 1018
13.I a game called “Trip To Jerusalem”, in how many ways can 7 people
be seated?
A. 270 B. 720 C. 5004 D. 5040
14.In how many ways can 8 teachers be seated in a circular conference?
A. 5040 B. 5400 C. 32 400 D. 40 320
15.In a group of 9 teachers, in how many ways can they sit in a circular
meeting?
A. 40 302 B. 40 320 C. 360 882 D. 362 880
Lesson
3
What I Need to Know
In this lesson, you will learn solving problems involving distinguishable
permutations. This are word problems related to real-life situation with
repeated symbols, and restrictions or special conditions. To further
understand the concept of solving problems involving permutations, use the
given formula as follow:
P = n!__
p!q!r!
12
Distinguishable
Permutations
20. What I Know
Pre-test
Solve the following problems using the concepts of distinguishable
permutations:
1. How many ways can we arrange the word “INFORMATION” so that all
the letters come together?
A. 2160 B. 4320 C. 360 984 D. 4 989 600
2. In Jhazelle’s bag, there are 3 books of Mathematics, 4 books of
English, and 2 books of Science. In how many ways can Jhazelle
arrange the books so that all the books of the same subjects are
together? A. 6 B. 9 C. 1260 D. 1728
3. Find the number of permutations of the word ALLAHABAD
A. 5 650 B. 6 750 C. 7 560 D. 7 650
4. How many ways can we arrange the letters of the word
MATHEMATICS?
A. 4 899 600 B. 4 986 900 C. 4 989 600 D. 6 898 400
5. Find the number of distinguishable permutations of the digits 122 838.
A. 180 B. 160 C. 18 D. 16
6. Find the number of permutations of the word COMMITTEE.
A. 45 630 B. 45 360 C. 43 530 D. 43 350
7. How many distinguishable permutations are possible in the word
SUCCESS? A. 1206 B. 1260 C. 1602 D. 1620
8. Find the number of permutations of the word COLLABORATION.
A. 259 200 459 C. 459 200 594
B. 259 459 200 D. 459 259 200
9. Find the number of permutations of the digits 203 224 304.
A. 5067 B. 7560 C. 7605 D. 7650
10. How many distinguishable permutations are possible in the word
AMPHITHEATER?
A. 27 600 993 C. 29 937 600
B. 29 600 937 D. 29 960 037
11.How manay different ways can the letters of the word BANANA be
arranged? A. 5 B. 6 C. 50 D. 60
12. Find the number of distinguishable permutations of the letters in
CALIFORNIA.
A. 200 907 B. 207 900 C. 907 020 D. 907 200
13. Find the number of distinguishable permutations of the letters in
INDEPENDENCE.
A. 1632006 B. 1663200 C. 2001636 D. 2016300
14. Find the number of distinguishable permutations of the given letters
“AAABBBCDDEEF”.
A. 2640033 B. 2640373 C. 3326400 D. 4002633
13
21. 15. How many ways can we arrange the word SUCCESS?
A. 204 B. 240 C. 402 D. 420
What’s In
This lesson emphasizes the distinct number of objects or elements in a given
set. So in solving distinguishable permutations, a need to consider
duplications of elements to make arrangements distinct.
What’s New
Activity 1:
1. Find the number of distinguishable permutation of the digits 120 022.
Solution:
Identify first the given
n = 6
0 - repeated twice (2)
2 - repeated thrice (3)
1 - once
P = n!_
p! q! r!
= _6!_
2! 3!
= 6.5.4.3.2.1
(2.1)(3.2.1)
= 6.5.4
2.1
= 120
2
P = 60 ways
Therefore, the number of distinguishable permutations of the digit 120
022 is equal to 60 ways.
14
22. What Is It
Activity 2:
Find the number of distinguishable permutations of the letters of the
word MISSISSIPPI.
Solution:
Identify first the given
n = 11
M - repeated once (1)
I - repeated four times (4)
S - repeated four times (4)
P - repeated twice (2)
P = n!_
p! q! r!
= _11!_
4! 2! 2!
= 11.10.9.8.7.6.5.4.3.2.1
(4.3.2.1)(2.1)(2.1)
= 11.10.9.8.7.6.5
(2.1)(2.1)
= 1,663,200
4
P = 415, 800 ways
Therefore, the number of distinguishable permutations of the word
MISSISSIPPI is equal to 415, 800 ways.
What’s More
Activity 3: Find Me!
Find the number of permutations of the following. Write all your answers on
your activity notebook:
1. BUKIDNON
2. MARAMAG
3. EARTHQUAKE
4. PHILIPPINES
5. ASSIGNMENTS
15
23. What I Haved Learned
Activity 4:
Find the number of distinguishable permutation of the following digits.
Write all your solution on your activity notebook:
1. 143 311
2. 120 000 450
3. 390 320
4. 645 456
5. 369 363 943
What I Can Do
Activity 5:
How many possible arrangements in the following words:
1. INFINITE
2. ARITHMETIC
3. INDEPENDENCE
4. ARCHITECTURE
5. ARCHEOLOGIST
Assessment
Solve the following problems using the concepts of distinguishable
permutations:
1. Find the number of distinguishable permutations of the given letters
“AAABBBCDDEEF”.
A. 2640033 B. 2640303 C. 3326400 D. 4002633
2. How many ways can we arrange the word SUCCESS?
A. 204 B. 240 C. 402 D. 420
3. How many ways can we arrange the word “INFORMATION” so that all
the letters come together?
A. 2160 B. 4320 C. 360 984 D. 4 989 600
4. Find the number of permutations of the word ALLAHABAD
A. 5 650 B. 6 750 C. 7 560 D. 7 650
5. How many ways can we arrange the letters of the word
MATHEMATICS?
A. 4 899 600 B. 4 986 900 C. 4 989 600 D. 6 898 400
16
24. 6. In Jhazelle’s bag, there are 3 books of Mathematics, 4 books of
English, and 2 books of Science. In how many ways can Jhazelle
arrange the books so that all the books of the same subjects are
together? A. 6 B. 9 C. 1260 D. 1728
7. How manay different ways can the letters of the word BANANA be
arranged? A. 5 B. 6 C. 50 D. 60
8. Find the number of distinguishable permutations of the digits 122 838.
A. 16 B.18 C. 160 D. 180
9. How many distinguishable permutations are possible in the word
SUCCESS? A. 1206 B. 1260 C. 1602 D. 1620
10. Find the number of permutations of the word COLLABORATION.
A. 259 200 459 C. 450 250 459
B. 259 459 200 D. 459 259 200
11.Find the number of permutations of the word COMMITTEE.
A. 43 350 B. 43 530 C. 45 360 D. 45 630
12.Find the number of permutations of the digits 203 224 304.
A. 5067 B. 7560 C. 7605 D. 7650
13. Find the number of distinguishable permutations of the letters in
CALIFORNIA.
A. 200 907 B. 207 900 C. 907 200 D. 970 260
14. How many distinguishable permutations are possible in the word
AMPHITHEATER?
A. 27 600 993 C. 29 937 600
B. 29 600 937 D. 29 960 037
15. Find the number of distinguishable permutations of the letters in
INDEPENDENCE.
A. 1632006 B. 1663200 C. 2001636 D. 2016300
Lesson
4
What I Need to Know
You have learned that permutation is the arrangement of objects wherein the
order is important. In this lesson, the order of the objects does not matter only
its arrangement. Solving problems involving combination of n objects taken r
17
Combination of n taken r at
a time
25. at a time is given emphasis. To further understand the concept of solving
problems involving combination, use the given formula as follow:
C = n!__
(n – r)! r!
What I Know
Pre-test
Solve the following problems using the concepts of combinations. Write your
answer on your activity notebook:
1. In serving a breakfast meal, in how many ways can Roland choose his
4 viand meal if there are 8 available dishes?
A. 24 B. 70 C. 720 D. 40 320
2. If ice cream is serve in a cone, in how many ways can Joshua choose
his three flavor ice cream scoop if there are 5 available flavors?
A. 6 B. 10 C. 120 D. 40 320
3. In how many ways can 5 teacher applicants be chosen from 100
qualified applicants?
A. 120 B. 288 C. 3 628 800 D. 75 287 520
4. In deciding 3 courses, in how many ways can Ivan choose out from 9
programs?
A. 6 B. 84 C. 120 D. 362 880
5. From 42 students of grade 10 students, how many ways can you
nominate 12 class officers?
A. 24 C. 11 058 116 888
B. 850 668 D. 47 129 001 600
6. In how many ways can a volleyball coach choose the first five players
from a group of 20 students?
A. 2 B. 120 C. 155 D. 15 504
7. Dave wants to make halo-halo. In how many ways he can choose 5
ingredients from a selection of 9 fruits?
A. 24 B. 120 C. 126 D. 362 880
8. In how many ways can 5 passengers be seated in a tricycle of there
are 6 available seats?
A. 1
B. 6
C. 120
D. 720
9. In how many ways a student choose 4 cake pops if there are 9 flavors
available?
A. 24 B. 125 C. 126 D. 362 880
18
26. 10. In how many ways can you choose 3 As and 2 Kings from a standard
deck of 52 cards?
A. 2 B. 6 C. 24 D. 120
Evaluate the following:
11. 10C4 14. 8C3
12. 6C2 15. 13C7
13. 20C14
What’s In
This lesson emphasizes the arrangement objects or elements in a given set.
We are only interested in the the number of groups or combinations. The
order of the elements is not important.
What’s New
Activity 1:
In how many ways the teacher formed 8 members from 25 students in
her section as class officers?
Solution:
Identify the given
n = 25 (Number of students)
r = 8 (students to be selected as class officers)
Apply the formula:
nCr = n!__
(n – r)! r!
25C8 = 25!__
(25 – 8)!8!
= 25!_
17! (8!)
= 25.24.23.22.21.20.19.18.17.16.15.14.13.12.11.10.9.8.7.6.5.4.3.2.1
(17.16.15.14.13.12.11.10.9.8.7.6.5.4.3.2.1)(8.7.6.5.4.3.2.1)
= 25.24.23.22.21.20.19.18
8.7.6.5.4.3.2.1
25C8 = 1 081 575 ways
Therefore, there are 1 081 575 ways if the teacher formed 8 members
from 25 students as class officers.
19
27. What Is It
Activity 2: Let’s have another example!
Find the possible combinations
1. C (10, 4)
Solution:
Identify the given
n = 10
r = 4
Apply the formula
nCr = n!__
(n – r)! r!
10C4 = 10!__
(10 – 4)!4!
= 10!_
6! (4!)
= 10.9.8.7.6.5.4.3.2.1
(6.5.4.3.2.1)(4.3.2.1)
= 10.9.8.7
4.3.2.1
10C4 = 210 ways
What’s More
Activity 3: Loosen Up!
Evaluate the following using the combination formula:
1. In how many ways can you nominate 10 class officers from 40 grade
10 students?
2. In how many ways can a basketball coach choose the first five players
from a group of 25 players?
3. From 100 qualified applicants, in how many ways can 2 teacher
applicants be chosen?
20
28. What I Haved Learned
Activity 4: Solve!
Evaluate the following using the combination formula:
1. If there are 7 ice cream flavors available, in how many ways Dave
choose 4 flavors?
2. In a jeepney there are 14 available seats. In how many ways can 6
passengers be seated?
3. How many ways can you choose 2 Jacks and 1 Queen from a standard
deck of 52 cards?
What I Can Do
Activity 5: Perfect Combination!
Solve the following:
1. 16C5
2. 29C13
3. 13C8
4. 100C7
5. 7C2
Assessment
Solve the following problems using the concepts of combinations. Write your
answer on your activity notebook:
1. In how many ways can you choose 3 As and 2 Kings from a standard
deck of 52 cards?
A. 2 B. 6 C. 24 D. 120
2. In serving a breakfast meal, in how many ways can Roland choose his
4 viand meal if there are 8 available dishes?
A. 24 B. 70 C. 720 D. 40 320
3. If ice cream is serve in a cone, in how many ways can Joshua choose
his three flavor ice cream scoop if there are 5 available flavors?
A. 6 B. 10 C. 120 D. 40 320
4. In how many ways can 5 teacher applicants be chosen from 100
qualified applicants?
A. 120 B. 288 C. 3 628 800 D. 75 287 520
5. In deciding 3 courses, in how many ways can Ivan choose out from 9
programs?
A. 6 B. 84 C. 120 D. 362 880
21
29. 6. In how many ways a student choose 4 cake pops if there are 9 flavors
available?
A. 24 B. 125 C. 126 D. 362 880
7. In how many ways can a volleyball coach choose the first five players
from a group of 20 students?
A. 2 B. 120 C. 155 D. 15 504
8. Dave wants to make halo-halo. In how many ways he can choose 5
ingredients from a selection of 9 fruits?
A. 24 B. 120 C. 126 D. 362 880
9. From 42 students of grade 10 students, how many ways can you
nominate 12 class officers?
A. 24 C. 11 058 116 888
B. 850 668 D. 47 129 001 600
10. In how many ways can 5 passengers be seated in a tricycle of there
are 6 available seats?
A. 1 B. 6 C. 120 D. 720
Evaluate the following:
11. 20C14 14. 6C2
12. 13C7 15. 10C4
13. 8C3
22
30. SUMMARY
This module was about solving problems involving permutations and combinations,
and its applications to real-life situations. Every lesson in this module was provided
with differerent advantageous chance to real-life problems which involve
permutations and combinations.
23
31. ASSESSEMENT
(Post-Test)
This time you are going to assess yourself about the things you have learned
in this module.
Direction: Read the following question below and encircle the letter of the
correct answer.
1. Find the number of distinguishable permutations of the letters of the
word COMMITTEE.
A. 9 B. 18 C. 40 320 D. 45 360
2. It is a way of selecting a subset from the given set where order does
not matter.
A. Combination C. Permutation
B. Integration D. Simulation
3. In how many ways can a committee of 9 students be chosen from 8
juniors and 10 seniors if there must be 5 seniors in the committee?
A. 190 B.920 C. 30 190 D. 31 920
4. In how many ways can you arrange 9 pots in a row?
A. 40 320 B. 42 300 C. 362 880 D. 380 620
5. In how many ways can 10 people be seated around a circular table?
A. 362 880 B. 368 028 C. 803 268 D. 862 286
6. In how many ways can n different objects be arranged taken all at a
time?
A. (n – 1)! B. (n + 1)! C. n! D. (2n)!
7. What is the arrangement of finite numbers of objects taken some or all
at a time?
A. Combination B. Permutation C. Probability D. Statistics
8. What is P(9,2)?
A. 2 B. 72 C. 40 320 D. 362 880
9. In a room, there are 8 chairs in a row. In how many ways can 6
students be seated in consecutive chairs?
A. 2 B. 720 C. 20 160 D. 21 060
10.A restaurant offers 4 kinds of soup, 7 kinds of main dish, 5 kinds of
vegetable dish, and 6 kinds of dessert. In how many possible ways can
the restaurant form a meal consisting of 1 soup, 2 main dish, 3
vegetable dish, and 2 desserts?
A. 8 B. 22 C. 106 D. 990
11. What is P(8, 3)?
A. 6 B. 336 C. 633 D. 40 320
12. In how many ways can 10 students arrange themselves in a
picture taking?
A. 362 800 B. 800 632 C. 3 628 800 D. 8 362 300
13. Find: C(9, 3)
A. 6 B. 84 C. 720 D. 362 880
14.Solve : C(10, 7) = _____.
A. 45 B. 25 C. 100 D. 120
15.Find: C(8, 3)
A. 36 B. 46 C. 56 D. 66
24
33. References
Cristobal, R. (2015). Math World 10. C & E Publishing Inc.
https://math.info/Algebra/Distinguishable Permutations
https://tinyurl.com/ybaskqs5
https://tinyurl.com/y77jn59q
Department of Education Mathematics 10 Learner’s Guide
Department of Education Mathematics 10 Teacher’s Guide
26
